Image Processing Reference
In-Depth Information
nal. This is precisely the equivalent to the output of the 3-point grayscale running
median applied to the original signal and implemented via rank ordering. A sorting
operation has thus been replaced by a counting operation which may in some cases
prove a useful property for more efficient implementation.
Filters that give the same answer through direct grayscale and stack filter imple-
mentation are said to obey the stacking property or to commute with thresholding.
For a filter to commute with thresholding, its binary version must be based on a
positive Boolean function (PBF)
. That is, the function must be capable of being
Figure 7.2
Stacking.
m
- 1 binary signals are transformed back to an
m
-level grayscale
waveform.
Figure 7.3
Stack filter implementation of a 3-point running median. The stack filter is imple-
mented via two separate routes. Starting at the top left the signal may be transformed into
three binary signals via threshold decomposition (bottom left). These are then individually
filtered using a 3-point running binary median, which is in effect a counting operation. The
resulting binary signals (bottom right) may then be stacked to produce the median filtered
grayscale signal (top right). This is precisely the same result achieved by carrying out a
3-point running grayscale median filter, based on rank ordering. The process of threshold
decomposition has thus replaced a sorting operation by three separate counting opera-
tions.
